1 Multiple Regression A single numerical response variable, Y. Multiple numerical explanatory variables, X 1, X 2,…, X k

2 Multiple Regression

3 Example Y, Response – Effectiveness score based on experienced teachers’ evaluations. Explanatory – Test 1, Test 2, Test 3, Test 4.

4 Test of Model Utility Is there any explanatory variable in the model that is helping to explain significant amounts of variation in the response?

5 Conclusion At least one of Test 1, Test 2, Test 3 or Test 4 is providing statistically significant information about the evaluation score. The model is useful. Maybe not the best, but useful.

6 Individual Slope Parameters In order to see what statistical tests of hypothesis for the various parameters in multiple regression model indicate, we need to go back to simple linear regression.

7 SLR – EVAL on Test 1 Predicted Eval = *Test1 For each additional point scored on Test 1, the Evaluation score increases by points, on average.

8 Explained Variation R 2 = 0.295, only 29.5% of the variation in Evaluation is explained by the linear relationship with Test 1.

9 Inference on t-Ratio = 2.97 P-value = Reject the null hypothesis that =0, because the P-value is so small. There is a statistically significant linear relationship between EVAL and Test 1.

10 Model with Test 1 If Test 1 is the only explanatory variable in the model, then the scores for Test 2, Test 3 and Test 4 are ignored by this model. What happens if we add Test 2 to the model with Test 1?

11

12 Model with Test 1, Test 2 Predicted EVAL = *Test *Test2 For each additional point on Test 1, while holding Test 2 constant, the Evaluation score increases by points, on average.

13 Model with Test 1, Test 2 Predicted EVAL = *Test *Test2 For each additional point on Test 2, while holding Test 1 constant, the Evaluation score increases by points, on average.

14 Explained Variation R 2 = 0.367, 36.7% of the variation in Evaluation is explained by the linear relationship with Test 1 and Test 2.

15 Explained Variation R 2 = – Test 1 and Test 2. R 2 = – Test 1 alone – = 0.072, 7.2% of the variation in EVAL is explained by the addition of Test 2 to Test 1.

16 Parameter Estimates – Test 2 Has Test 2 added significantly to the relationship between Test 1 and Evaluation? Note that this is different from asking if Test 2 is linearly related to Evaluation!

17 Parameter Estimates – Test 2 t-Ratio = 1.51 P-value = Because the P-value is not small, Test 2’s addition to the model with Test 1 is not statistically significant.

18 Parameter Estimates – Test 2 Although R 2 has increased by adding Test 2, that increase could have happened just by chance. The increase is not large enough to be deemed statistically significant.

19 Parameter Estimates – Test 1 Does Test 1 add significantly to the relationship between Test 2 and Evaluation? Note that this is different from asking if Test 1 is linearly related to Evaluation!

20 Parameter Estimates – Test 1 t-Ratio = 2.52 P-value = Because the P-value is small, Test 1’s addition to the model with Test 2 is statistically significant.

21 Parameter Estimates – Test 1 If we had started with a model relating Test 2 to EVAL, adding Test 1 would result in an increase in R 2. That increase is large enough to be deemed statistically significant.